# Relation between marginal cost and output elasticities

Does anyone know of any results that show the link between marginal cost and the output elasticities analytically? I am looking at production and cost theory books but can't find any results that posit a direct dependence.

• Why do you think that marginal costs are a function of the $sum$ of output elasticities? Mar 25 '21 at 13:41
• I'm saying the sum because the sum of output elasticities = the returns to scale, which are related to marginal cost, as I say in the first sentence. Mar 26 '21 at 21:43
• Edited the question to remove the part relating to the returns to scale, as it was confusing. Apr 6 '21 at 21:04

I figured it out:

The first-order condition of the cost minimization problem for, say, material inputs $$m_{it}$$ gives:

$$\lambda \frac{\partial F}{\partial M} = P_M$$

Where F is the production function, $$P_M$$ the material input prices. Multiply by $$\frac{M}{F}$$ and rearrange,

$$\lambda = \frac{P_M M}{\beta_M F}$$,

where $$\beta_M$$ is the output elasticity with respect to material inputs. Similar results obtain for other inputs. Summing them all gives the relation to the short-run returns to scale.

• It should be $\lambda = \frac{P_M M}{\beta_M F}$, and in the end it returns a formula for marginal cost that is not very useful. Specifically, it does not return marginal cost as a function of the sum of elasticities and therefore does not establish a connection to the returns to scale. There is no such connection, since by scaling up all inputs you typically violate cost minimization. Mar 29 '21 at 12:52
• You're right, thanks. Typo in the formula, and one can only get the short-run returns to scale. Corrected. Mar 29 '21 at 21:10
• I don't agree with the "not very useful", since I was mostly looking for a relation between MC and output elasticities. Mar 29 '21 at 21:11
• Hmm, but it was formulated as if you were looking for a relation between MC and the sum of output elasticities. (That's why I asked about that on March 25.) I'm also skeptical about the "short-run returns to scale" - what's that? I still believe the formula is not very useful, since the elasticities alone don't help, as you also need to know the expansion path $M_i(F)$ to calculate MC for given output and prices. Mar 30 '21 at 11:49
• Yeah, let's close it before the mods step in. Apr 7 '21 at 7:31